Let X = {1, 2, 3, 4, 5}. The number of different ordered pairs (Y, Z) that can be formed such that Y ⊆ X, Z ⊆ X and Y ∩ Z is empty, is

• 5²

• 3⁵

• 2⁵

• 2⁵

Let R be the set of real numbers.
Statement-1 : A = {(x, y) ∈R × R : y - x is an integer} is an equivalence relation on R.
Statement-2 : B = {(x, y) ∈ R × R : x = αy for some rational number α} is an equivalence relation on R.

• Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1. 

• (2) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1. 

• Statement-1 is true, Statement-2 is false. 

• Statement-1 is true, Statement-2 is false. 

If A, B and C are three sets such that A ∩ B = A∩ C and A ∪ B = A ∪ C, then

• A = B

• A = C

• B = C

• B = C

A die is thrown. Let A be the event that the number obtained is greater than 3. Let B be the event that the number obtained is less than 5. Then P (A ∪ B) is 

• 3/5

• 0

• 1

• 1

Let A and B be two events such that  where  stands for complement of event A. Then events A and B are

• equally likely and mutually exclusive

• equally likely but not independent

• independent but not equally likely

• independent but not equally likely

Two sets A and B are as under:

A = {(a-b)∈ RxR:|a-5|<1 and |b-5|<1}

B = {(a,b)∈ Rx R: 4(a-6)² + 9 (b-5)² ≤ 36},then

• Neither A ⊂ B nor B ⊂ A

• B ⊂ A

• A ⊂ B

• A ∩ B = ∅

On R, the set of real numbers, a relation p is defined as 'a$\mathrm{\rho }$b if and only if 1+ ab> 0'. Tnen,

• $\mathrm{\rho }$ is an equivalence relation

• $\mathrm{\rho }$ is reflexive and transitive but not symmetric 

• $\mathrm{\rho }$ is reflexive and symmetric but not transitive 

• $\mathrm{\rho }$ is only symmetric

Let R be a relation defined on the set Z of all integers and xRy, when x + 2y is divisible by 3, then

• A is not transitive

• R is symmetric only

• R is an equivalence relation

• R is not an equivalence relation

189.

Let f : N $\to$ R be such that f(1) = 1 and f(1) + 2f(2) +  3f(3) + ... + nf(n) = n(n+ 1) f(n), for all n  $\in$ N, n $\ge$ 2, where N is the set of natural numbers and R is the set of real numbers. Then, the value of f(500) is

• 1000

• 500

• 1/500

• 1/1000

For the function f(x) = $\left[\frac{1}{\left[\mathrm{x}\right]}\right]$. where [x] denotes the greatest integer less than or equal to x, which of the following statements are true?

• The domain is $\left(- \infty , \infty \right)$

• The range is $\left\{0\right\} \cup \left\{- 1\right\} \cup \left\{1\right\}$

• The domain is $\left(- \infty , 0\right) \cup [1, \infty )$

• The range is $\left\{0\right\} \cup \left\{1\right\}$